Optimal. Leaf size=67 \[ \frac{2 (d x)^{5/2} \left (\frac{b x^2}{a}+1\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (\frac{5}{4},-2 p;\frac{9}{4};-\frac{b x^2}{a}\right )}{5 d} \]
[Out]
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Rubi [A] time = 0.0629362, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{2 (d x)^{5/2} \left (\frac{b x^2}{a}+1\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (\frac{5}{4},-2 p;\frac{9}{4};-\frac{b x^2}{a}\right )}{5 d} \]
Antiderivative was successfully verified.
[In] Int[(d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^p,x]
[Out]
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Rubi in Sympy [A] time = 17.0761, size = 60, normalized size = 0.9 \[ \frac{2 \left (d x\right )^{\frac{5}{2}} \left (1 + \frac{b x^{2}}{a}\right )^{- 2 p} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - 2 p, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{5 d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x)**(3/2)*(b**2*x**4+2*a*b*x**2+a**2)**p,x)
[Out]
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Mathematica [A] time = 0.0256822, size = 56, normalized size = 0.84 \[ \frac{2}{5} x (d x)^{3/2} \left (\left (a+b x^2\right )^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-2 p} \, _2F_1\left (\frac{5}{4},-2 p;\frac{9}{4};-\frac{b x^2}{a}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^p,x]
[Out]
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Maple [F] time = 0.017, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{{\frac{3}{2}}} \left ({b}^{2}{x}^{4}+2\,ab{x}^{2}+{a}^{2} \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x)^(3/2)*(b^2*x^4+2*a*b*x^2+a^2)^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \left (d x\right )^{\frac{3}{2}}{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)^(3/2)*(b^2*x^4 + 2*a*b*x^2 + a^2)^p,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{d x}{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} d x, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)^(3/2)*(b^2*x^4 + 2*a*b*x^2 + a^2)^p,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)**(3/2)*(b**2*x**4+2*a*b*x**2+a**2)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \left (d x\right )^{\frac{3}{2}}{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)^(3/2)*(b^2*x^4 + 2*a*b*x^2 + a^2)^p,x, algorithm="giac")
[Out]